Solve the differential equation $$y′= 6x^2\sqrt y $$ and draw the phase portrait. Find out how many solutions with initial conditions $(x_0,y_0) = (a,b)$ this equations possesses depending on $(a,b)$. Consider both cases: local and global solutions.
I have solved the differential equation and got $y = (x^3-c)^2$, but i dont know how to do the rest of the question. Also is it importent to let it be $-c$ instead of $+c$.
Use the site indicated at the top of this image which will give you a graphical representation at a certain scale of the field of "speed vectors" with coordiantes $(x,y')$ but also an idea of any solution if you provide initial data.
Do you see how well the solution curve I have asked follows the vector field ?
It is indeed necessary to assume $c<0$.